In objectivism, morality (principles for action) have an OBJECTIVE basis (life as the standard of value) and would never be based on either the whims of others, or the whims of ones' self projected onto others.... which is what the Golden Rule would have us make the basis for moral determinations.
I think the contradiction between that and the Golden Rule is pretty plain to see.

"He thought of all the living species that train their young in the art of survival, the cats who teach their kittens to hunt, the birds who spend such strident effort on teaching their fledglings to fly—yet man, whose tool of survival is the mind, does not merely fail to teach a child to think, but devotes the child’s education to the purpose of destroying his brain, of convincing him that thought is futile and evil, before he has started to think. From the first catch-phrases flung at a chi...

My principle objection is not with the core of Rand's philosophy but the failure to integrate her ethics with it's inevitable implications about politics. I think that her ethics renders all government (coercive monopoly on protection, adjudication of law) immoral to the extent that it is maintaned by initiation of force to either fund or maintain the status of single provider. A voluntarily-funded government as she proposed, which also did not initiate force to maintain its monopoly, in effect is no longer a government but at that point is another provider of services in a free, competitive market. I've yet to hear a convincing refutation of this determination, and it's important to me because I'm not just an anarcho-capitalist looking for holes to poke in Rand's legacy. I'm very much an objectivist under the core philosophy and it was Atlas Shrugged that actually pushed me away from minimal-state libertarianism and to full repudiation of the notion of government.